The concept of topological matter has been generalized to non-Hermitian systems in recent years (see Refs. [1]}, [2]}, [3]}, [4]}, [5]} for reviews).
In the presence of gain and loss or nonreciprocal effects, unique topological phenomena without
any counterparts in closed systems could emerge, such as the non-Hermitian skin effect (NHSE) [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} and exceptional topological phases [5]}.
The interplay between time-periodic drivings and non-Hermitian effects could further induce
intriguing phases of matter in out-of-equilibrium situations, like the non-Hermitian Floquet topological
insulators [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, superconductors [26]}, [27]}, semimetals [28]}, [29]}, [30]}, [31]} and quasicrystals [32]}, [33]}, [34]}. As reported in our findings below, applying our \(q\) th-rooting procedure to such periodically driven non-Hermitian Floquet topological phases yields even more exotic features absent in their original counterpart, such as fractional-quasienergy topological edge or corner modes.
